Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and dire

Identify the coordinates of the vertex and focus, the...

Key Concepts

Latus Rectum

The latus rectum is a line segment that passes through the focus and is perpendicular to the axis of symmetry. Its length is directly related to the geometric properties of the parabola, specifically influencing its 'width' at the focus, and it serves as a measure for the distance between the two points where a line parallel to the directrix intersects the parabola.

Directrix

The directrix of a parabola is a fixed line used in its definition. For any point on the parabola, the distance to the focus equals the perpendicular distance to the directrix. This concept is central to deriving the equation of a parabola and understanding its geometric construction.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical or horizontal line (depending on the orientation) that passes through the vertex and divides the parabola into two mirror-image halves. This concept simplifies the process of graphing the parabola and helps in understanding its geometric properties.

Direction of Opening

The direction of opening of a parabola determines whether it opens upward, downward, to the right, or to the left. This is determined by the sign and position of the squared term in the equation, and it has significant implications for the set of solutions and the graphical representation of the parabola.

Vertex

The vertex of a parabola is the point where the curve changes direction; it is the minimum or maximum point of the parabola depending on its orientation. In the standard form of a parabolic equation, the vertex is easily identifiable, and it serves as a key reference point for graphing the parabola as well as determining its axis of symmetry.

Parabola (Conic Section)

A parabola is a set of all points equidistant from a fixed point, called the focus, and a fixed line, known as the directrix. It is one of the conic sections, formed by the intersection of a plane and a right circular cone, and has unique reflective and geometric properties that are widely used in physics and engineering.

Focus

The focus is a fixed point used in the definition of a parabola. Each point on the parabola is equidistant from the focus and the directrix. Knowledge of the focus is essential for understanding the reflective property of parabolas, and it is key in constructions and applications involving parabolic mirrors or antennas.

Transcript

00:01 For the equation of the parabola, y plus 2 equals negative x minus 3 squared.

00:06 Find the coordinates of the vertex and focus, the equations of the directrix and axis of symmetry, and the direction of the opening.

00:15 Then find the length of the lattice rectum and graph the parabola.

00:19 Well, to start off, i'm going to put this equation in standard form, and that would, i'd need to move the two, i need to subtract two from both sides, so i get y equals, and i'm going to write this as negative 1, just to make it easier to see that a is negative 1, x minus 3 squared minus 2.

00:44 So in this equation, i've got a is negative 1, h, small h, is 3, k is negative 2.

00:56 So the different things i need to find.

00:59 First of all the vertex the vertex is hk so the vertex is going to be 3 negative 2 next for this equation since it's y equals and we have the x squared the focus is h so that's going to be 3 and then k plus 1 over 4a so that's going to be negative 2 plus 1 over 4 times negative 1.

01:36 So that's negative 2 plus negative 1.

01:38 So that's negative 2 plus negative a 4th.

01:42 So that focus is 3 and then negative 2 and 1 4th.

01:49 Now let's find our equations.

01:52 And first of all, for the directrix, the equation is y equals k minus 1 over 4a.

01:58 So that's going to be y equals, well here was k plus 1 over 4a so instead of negative 2 minus a 4th it's going to be negative 2 plus a 4th so that's negative 1 and 3 4th so that's our directrix and then for the axis of symmetry the equation would be x equals h so that's the equation x equals 3 that's simple enough the direction of the opening, well again since it's y equals and x squared it's either going up or down and since a is negative the direction of the opening is down so there we have it.

02:49 All right now we're going to find the length of the lattice rectum and the length of the lattice rectum is the absolute value of one over a so that's the absolute value of one over negative one which is the absolute value of one over negative one which is the absolute value of one...

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